Optimal. Leaf size=44 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{\sqrt{10}-2}} x\right ),\frac{1}{3} \left (2 \sqrt{10}-7\right )\right )}{\sqrt{2+\sqrt{10}}} \]
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Rubi [A] time = 0.0679591, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-2+\sqrt{10}}} x\right )|\frac{1}{3} \left (-7+2 \sqrt{10}\right )\right )}{\sqrt{2+\sqrt{10}}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-4 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{-4+2 \sqrt{10}-4 x^2} \sqrt{4+2 \sqrt{10}+4 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-2+\sqrt{10}}} x\right )|\frac{1}{3} \left (-7+2 \sqrt{10}\right )\right )}{\sqrt{2+\sqrt{10}}}\\ \end{align*}
Mathematica [C] time = 0.0536968, size = 51, normalized size = 1.16 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{2+\sqrt{10}}} x\right ),-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right )}{\sqrt{\sqrt{10}-2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.209, size = 84, normalized size = 1.9 \begin{align*} 3\,{\frac{\sqrt{1- \left ( 2/3+1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( 2/3-1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{6+3\,\sqrt{10}},i/3\sqrt{15}-i/3\sqrt{6} \right ) }{\sqrt{6+3\,\sqrt{10}}\sqrt{-2\,{x}^{4}-4\,{x}^{2}+3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} - 4 \, x^{2} + 3}}{2 \, x^{4} + 4 \, x^{2} - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{4} - 4 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} - 4 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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